Praise be to Allah.
The hadeeth referred to in the question is saheeh. It was narrated by Muslim in his Saheeh (2132) from the hadeeth of Ibn ‘Umar (may Allah be pleased with him) who said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “The most beloved of your names to Allah are ‘Abd-Allah and ‘Abd al-Rahmaan.”
Abu Dawood (4950) narrated that Abu Wahb al-Jushami, who was a companion of the Prophet (peace and blessings of Allah be upon him), said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Give your children the names of Prophets, and the most beloved of names to Allah are ‘Abd-Allah and ‘Abd al-Rahmaan, and the most truthful are Haarith and Hammaam [these names both refer to one who is always doing something, whether good or bad], and the worst are Harb and Murrah [meaning ‘war’ and ‘bitterness’ respectively].” Classed as saheeh by al-Albaani in Saheeh Abi Dawood.
The names which reflect servitude to Allah (those that start with “ ‘Abd” meaning “slave of”) are more beloved to Allah than others, because they affirm the attributes that befit Him and do not befit anyone else. No one among mankind has any right to them or any share in them. This has to do with His divine Attributes (sifaat al-uloohiyyah). They also affirm the status that befits the person, which never alters for an instant throughout his lifetime, namely the status of servitude, of being the slave of his Lord. So the name reflects well on the one who carries it; it honours the person who is given this name by describing him as a slave of his Lord. Hence these names carry great virtue.
The connection between the slave and Allah is one of pure servitude (‘uboodiyyah), and the connection between Allah and the slave is one of complete mercy. It is by His mercy that he exists in the true sense of the word, and the purpose for which he was created is to worship Allah alone, with love, fear, hope, worship and veneration, so he is a slave to Allah.
This is what is meant by the name: that the owner of this name is connected to one of the attributes of Allah which cannot be attributed to anyone else.
Because His mercy prevails over His wrath, mercy is more beloved to Him than anger, so the name ‘Abd al-Rahmaan (“slave of the Most Merciful”) is dearer to him than the name ‘Abd al-Qaahir (“slave of the Subduer”) and similar names.
So these names refer to these two meanings, that of divinity and that of servitude, and that means that the name is a constant reminder of the slave’s humble position before his Lord, and a constant prayer for the Lord’s mercy towards His poor, humble servant.
Some scholars have stated that these two names have their own special features, which is that in the Qur’aan, Allah does not mention the word ‘abd (or its plural ‘ibaad) in conjunction with any of the names of Allah except these two. Allah says (interpretation of the meaning):
“And when the slave of Allah [‘abd Allah] (Muhammad) stood up invoking Him”
“And the (faithful) slaves of the Most Gracious [‘ibaad al-Rahmaan] (Allah)”
This is supported by the verse in which Allah says (interpretation of the meaning):
“Say (O Muhammad): Invoke Allah or invoke the Most Gracious [al-Rahmaan] (Allah)”
If the first thing of which a child becomes aware and understands and hears is this name, the idea will take root in his heart that he is a slave of Allah and that Allah is his Lord, and that his Lord is the Most Gracious, Most Merciful, and it will be easy to raise him accordingly.
For more information please see Tuhfat al-Mawdood bi Ahkaam al-Mawlood, by Ibn al-Qayyim; Fath al-Baari by Ibn Hajar, in his commentary on the chapter Ahabb al-Asma’ ila Allah (the most beloved names to Allah); and al-Sindi’s commentary on Ibn Maajah (Haashiyat al-Sindi ‘ala Ibn Maajah).
May Allah help us and you to be sincere slaves to Him, and admit us to His Mercy by His grace and bounty, for He is the most merciful of those who show mercy.